Bill Murphy, an 80-year-old retired veterinarian in suburban Phoenix, sometimes blanks on names he could once summon with ease, so he has empathy for 81-year-old President Biden. But he winced when he watched Mr. Biden defend his mental sharpness at a news conference, only to mix up the presidents of Egypt and Mexico. Mr. Murphy, a Republican, believes Mr. Biden is not up to another term.

Mary Meyer, an 83-year-old avid hiker and traveler who lives in the high desert north of Phoenix, took issue with a special counsel’s report that characterized him as elderly and forgetful — a similar assumption that strangers at the supermarket sometimes make about her capabilities.

“I look at him as a peer,” said Ms. Meyer, who plans to vote for Mr. Biden. “I know what he’s capable of. I know it’s not as bad as everybody thinks.”

To voters in their 70s and 80s, the renewed questions swirling around Mr. Biden’s age and fitness resonated in deeply personal ways. The special counsel report cleared him of criminal charges in his handling of classified documents but described him as a “sympathetic, well-meaning, elderly man with a poor memory.”

Some of Mr. Biden’s generational peers and supporters insisted the characterization was nothing more than a calculated political ploy to undercut his campaign, and play on perceived weakness. Many noted their own vibrant and busy lives, filled with mental and physical activity.

The criticism of Mr. Biden as forgetful and incapable of serving echoed slights and discrimination they had felt. Others thought of their own struggles as they hit their 80s, and questioned any 80-year-old’s ability to lead the nation.